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\begin{document}

\title{Notes about dynamic and thermodynamic stellar stability}
\author{G. S. Bisnovatyi-Kogan and S. I. Blinnikov}
\date{}

\begin{titlepage}

\maketitle
\thispagestyle{empty}
\begin{abstract}
X-ray properties of
Cygnus X-
\end{abstract}

\end{titlepage}
% \begin{center}

\section{Introduction}

Dynamic stability of stars without internal energy sources, like white dwarfs or neutron stars,
is defined  relative to perturbations with conserving entropy and specific angular momentum, and
is valid for a short time scale. Thermodynamic, or secular instability is related to perturbations, 
which try to return the system to a global thermodynamic equilibrium, which is characterized by
a constant temperature, and rigid rotation. The classic example is a dynamic and secular stability of
the rotating gravitating spheroid of the incompressible liquid, relative to the transformation into
 the 3-D ellipsoid (Chandrasekhar, book). The secular instability, at which there is an exchange of 
 angular momentum between layers, sets on well before the dynamic one.
 
 Dynamic stability of rigidly rotating stars was investigated by Bisnovatyi-Kogan and Blinnikov (1973), 
 using the static criteria of stability (Zeldovich, 1963).  
 It was shown that on the equilibrium curve of a mass $M$ versus a central density $\rho_c$, for rigidly 
 rotating isentropic stars the loss of a dynamic stability happens always after the maximum of the curve 
 $M(\rho_c)$, corresponding to rigidly rotating models with the same specific angular momentum, 
 see Fig.1. Here we show, that the maximum of this curve corresponds actually to the
 point of the loss of thermodynamic stability, if we include into the consideration the viscosity and 
 angular momentum exchange between layers. After this maximum the the star will become unstable and collapse,
 but the time of a development of this instability should be much larger than the dynamic time scale
 $t_d\sim (4\pi G\rho)^{-1/2}$, and is defined by the time of the viscous exchange of the angular momentum,
 connected with a matter or turbulent viscosity. Similar idea was developed by Friedman, Ipser and Sorkin (1988).
 
 Dynamic stability of the non-rotating isothermal white dwarfs was investigated by Bisnovatyi-Kogan (1966). 
 Using the same static criteria of stability, it was obtained that on the the loss of dynamic stability 
 occurs after the maximum of the curve  $M(\rho_c)$, corresponding to isothermal stars with a constant 
 temperature. If we take into account the thermal conductivity, the model will loose the stability exactly in 
 the maximum of this curve, but the time of the development of this secular instability will be defined by the
 heat exchange time between stellar layers by thermal conductivity.  
   
     

%{References}
\bigskip
\hrule
\bigskip

1. N. I. Shakura, ``A disk model for accretion of gas by a relativistic
star in a close binary system",
Astron. Zh. {\bf 49}, 921-929 (1972) [Sov. Astron.{\bf 16},756-762 (1973)].

2. J. E. Pringle and M.J.Rees, ``Accretion disk models
for compact X-ray sources", Astron. Astrophys. {\bf 21},
1-9 (1972).

3. N. I. Shakura and R. A. Syunyaev, ``Black holes in binary systems: observational appearance", Astron. Astrophys. {\bf 24}, 337-355 (1973).

4. I. D. Novikov and K. S. Thorne, ``Black-hole astrophysics",
 in: Black Holes (Les Houches lectures, Aug. 1972), Gordon and Breach (1973), pp.343-450.

5. F. Frontera and F. Fuligni, ``Energy-spectrum variability
 of Cyg X-1 in hard X-rays", Astrophys. J. {\bf 196},
597-599 (1975).

6. G. F. Carpenter, M. J. Coe, A. R. Engel, and J. J. Quenby, ``Ariel 5 hard X-ray measurements of galactic and extragalactic source spectra", Proc.14th Intl. Cosmic Ray Conf. (Munich) {\bf 1}, 174-179 (1975).

7. R. C. Haymes and F. L. Harnden, ``Low-energy $y$ radiation
from Cygnus" Astrophys. J. {\bf 159}, 1111-1114 (1970).

8. K. E. Baker, R. L. Lovett, K. J. Orford, and D. Ramsden,
``Gamma rays of 1-10 MeV from the Crab and Cygnus regions",
Nature Phys. Sci. {\bf 245}, 18-19 (1973).

9. K. S. Thorne and R. H. Price,
``Cyg X-1: an interpretation of the
spectrum and its variability", Astrophys. J. {\bf 195},
 L101-L105 (1975).

10. R. H. Price and E. P. Liang, Preprint (1975).

\noindent
[published in Astrophys. J., {\bf 218}, Nov. 15, 1977, 247-252.]

11. S. L. Shapiro, A. P. Lightman, and D. M. Eardley,
``A two-temperature accretion-disk model for Cyg X-l",
Astrophys. J. {\bf 204}, 187-199 (1976).

12. G. S. Bisnovatyi-Kogan and S. I. Blinnikov,
Preprint Inst. Kosmich. Issled.
Akad. Nauk SSSR No. 271 (1976); Astron. Astrophys. (in press).

\noindent
[published in A\&A 1977, {\bf 59}, 111-125]

13. M. Schwarzschild,
Structure and Evolution of the Stars, Princeton Univ.
Press. (1958).

14. L. Biermann and R. Lust. ``Nonthermal phenomena in
stellar atmospheres",
in: Stellar Atmospheres, Univ. Chicago Press (1960), Chap. 6.

15. D. R. Parsignault, A. Epstein, J. E. Grindlay,
 E. J. Schreier, H. Schnopper,
H. Gursky, Y. Tanaka, A. C. Brinkman, J. Heise, 1. Schrijver, R. Mewe,
E. Gronenschild, and A. den Boggende, ``ANS observations of Cyg X-1",
Astrophys. Space Sci. {\bf 42}, 175-184 (1976).

16. S. S. Holt, E. A. Boldt, L. J. Kaluzienski, and P. J. Serlemitsos,
``Observations of a new transition in the emission
from Cyg X-1", Nature {\bf 256}, 108-109 (1975).

17. L. A. Pustil'nik and V. F. Shvartsman,
``Possible influence of magnetic fields on the structure of a plasma
accretion disk in binary systems", in:
Gravitational Radiation and Gravitational Collapse (IAU Sympos. No. 64),
Reidel (1974), p. 213.

18. D. M. Eardley and A. P. Lightman,
``Magnetic viscosity in relativistic
 accretion disks", Astrophys. J. {\bf 200}, 187-203 (1975).

19. G. S. Bisnovatyi-Kogan and A. A. Ruzmaikin,
``The accretion of matter
by a collapsing star in the presence of a magnetic field",
Astrophys. Space Sci. {\bf 42}, 401-424 (1976).

20. P. Goldreich and W. H. Julian, ``Pulsar electrodynamics",
Astrophys. J., {\bf 157}, 869-880 (1969).


\newpage
\vglue 1cm
\begin{center}
{\Large {\bf Models for the X-ray brightness fluctuations in
Cygnus X-1 and active galaxy nuclei}
\bigskip

G. S. Bisnovatyi-Kogan and S. I. Blinnikov}
\bigskip

{\em Institute for Space Research, USSR Academy of Sciences, Moscow}

(Submitted April 7, 1978)

Pis'ma Astron. Zh. 4, 540-543 (December 1978)

Sov.Astron.Lett. 4, 290-291 (Nov.-Dec. 1978)
\end{center}

\begin{small}
The X-ray brightness fluctuations induced in
Cyg X-l and active galaxy nuclei by
convection and turbulence in the
sub-photospheric layers are discussed in terms of
the disk-accretion and supermassive-star
models. The variability time scales should be
comparable with those observed, but in the case of
supermassive stars and disks it is difficult to obtain the
observed amplitude of the fluctuations.


PACS numbers: 98.70.Qy, 98.50.Rn, 97.10.Cv
\end{small}

\bigskip

1. One noteworthy property of the X-ray source Cygnus X-l is the
variability of its flux on time scales ranging from milliseconds to a
fraction of a second [1].
This variability is generally attributed to the
presence of a turbulent accretion disk in the Cyg X-l system. In
particular, the flux variations have been explained by the rotation of a
hot spot [2,1] or alternatively, by the onset of instability in a zone where
radiation pressure predominates [3,5].

In this letter we shall consider another model for the variability.
We shall outline more fully what properties of the flux variations are to
be expected on the accretion disk model as a direct result of the
presence of turbulence, and how convective instability should develop
in a region of high radiation pressure [6,7]. Acoustic waves generated in
the convection zone will escape into optically thin layers, and will not
only induce variable soft X-rays in the photosphere, but will also be
responsible for variable heating of the corona. Comptonization of the
photospheric radiation by the hot electrons under conditions where both
temperature and density are variable will lead to flux variations in the
hard range, ${h\nu} \ga $ 5 keV. This X-ray fluctuation mechanism, involving
the emergence of waves into transparent layers, evidently is of the same
wave nature as the variability of the ultraviolet excess in stars
experiencing intensive convection, such as T Tauri and UV Ceti.

2. The waves escaping into the transparent layers and producing
variable radiation in the photosphere and corona will occupy a rather
narrow frequency band. Physically, the reason for this circumstance is
that media with a high radiation pressure and a nonuniform distribution
of plasma along the $z$ coordinate (across the disk) will serve as efficient
filters, isolating a characteristic frequency range from the broader
spectrum generated by convection and turbulence. Waves of low
frequency and a wavelength exceeding the scale height of the
atmosphere will not escape outside but will induce oscillations of
the coronal atmosphere as a whole. On the other hand, under conditions
where radiation pressure predominates, high-frequency waves will
experience very severe damping because of radiative friction, and their
role in heating the corona will be insignificant. High frequencies may,
however, appear in the observations either through nonlinear
transformation of low-frequency waves or due to inadequate processing of
the observational data [8].

We have examined elsewhere [9] the propagation of waves through
a medium with strong radiation pressure, followed by their escape into
the atmosphere. Waves emerging into the transparent layers will have a
phase and group velocity equal to the velocity of sound in gas,
$v_g = (\gamma P_g /\rho)^{1/2} = (\gamma{\cal R}T)^{1/2}$,
where $\gamma = $5/3 or $\gamma =$ 1 according as scattering
or absorption predominates, ${\cal R}$ is the gas constant,
and the temperature  $T$  of the equilibrium atmosphere is
approximately equal to the temperature $T_{\rm e}$ of the photosphere.
The characteristic frequency $\omega_{\rm c}$ of waves
emerging into the atmosphere is given by the expression
\begin{equation}
\omega_{\rm c}= \left(\frac{\gamma}{{\cal R}T}\right)^{1/2}
 g \left(1 - \frac{H}{H_c}\right) {\rm sec}^{-1} .
\end{equation}
For the accretion disk model, the gravitational acceleration $g$ at radius
$r$ is equal [10] to
\begin{equation}
 g = \frac{GM}{r^2} \frac{h_0}{r} ,
\end{equation}
where the characteristic thickness is
\begin{equation}
 h_0 = 3\frac{L}{L_c} R_0\left[1-\left (\frac{R_0}{r}\right)^{1/2}\right] ,
\end{equation}
and $R_0$ is the radius of the inner edge of the disk. The quantity
$H/H_{\rm c} = \kappa_0 H/gc$  represents the ratio of the
 radiation pressure force to the
gravity; $\kappa_0$ is the opacity, including both absorption and
scattering [9].
In a spherically symmetric star of luminosity $L$ and radius $R$ we will
have
$g = GM/R^2$, $H = L/4 \pi R^2$, and $H/H_c = L/L_c$,
where $L_c = 4 \pi cGM/\kappa_0$
is the Eddington limiting luminosity.

3. To calculate the characteristic frequencies of the fluctuations
for Cyg X-l, we shall adopt the convective accretion-disk models given
in a previous paper [7] and by Shakura et al. [11].
For a black hole of mass $M = 10 M_{\odot}$, we find that in
both models as the luminosity rises from $0.1 L_c$ to $0.3 L_c$
there is little change in the characteristic frequency $\omega_c$
corresponding to the equilibrium temperature, as given by Eq. (1); the
values of $\omega_{\rm c}$ are confined [9] to the range 5-40 msec.
Note that if a corona
with $T_c \approx 10^2 T_e$ is present,
waves whose frequency is $\omega = \omega_{\rm c}$ or even
somewhat lower will be able to escape.
The frequency range mentioned
is in good accord with the observed time-scales of variability.
Our mechanism can yield fluctuations weakly
correlated in time, and can simulate the white noise derived from
observational analysis [8] of the brightness fluctuations of Cyg X-l.

Despite the good agreement between the theoretical time
characteristics and the observations, analysis of small oscillations in
our model still does not suffice to explain the amplitudes of the
variability we observe, because pulsations should, in general, occur
independently in regions whose size is comparable with the velocity of
sound multiplied by the pulsation period, or about one-tenth the
diameter of the zone of maximum energy production. Small pulsations
should accordingly be smoothed out. However, the mechanism we are
proposing could operate in a highly nonlinear regime. Strong
nonlinearity of the waves would be expected in the light of theoretical
estimates [7], and is essential to the very existence of the corona. The
frequencies obtained from linear analysis correspond to the
characteristic growth times of strong nonlinear flares. This problem
awaits further theoretical treatment. Indeed, we would point out that
not even the observational situation is fully clear [8].

An evaluation of the proportion $\eta$ of the acoustic flux that
emerges into the corona and produces heating shows [9] that as the
luminosity rises by a factor of 3 from 0.1 $L_c$ to 0.3 $L_c$,
 the value of $\eta$
will drop by a factor of 5-10. Thus, in accord with our previous
suggestion [6,7], as the luminosity increases the power of the corona will
remain approximately the same, or may even decline somewhat. This
situation will prevent the hard X-ray flux
($h\nu > 5$ keV) of Cyg X-l from
changing appreciably as the total luminosity varies, in agreement with
the observations [12].

4. Multicolor photometry of the nuclei of certain galaxies has
revealed the presence of comparatively short-period brightness
fluctuations in the nucleus of the Seyfert galaxy [13,14] NGC 4151
($\approx$ 130 days) and in the BL Lacertae object [15]
 OJ 287 ($\approx$ 180 days). There is
presently no consensus as to the nature of active galaxy and quasar
nuclei. Three models have been explored [16,17]:
a) a dense star cluster; b) a supermassive star; c) disk accretion onto a
massive black hole. The variability mechanism proposed here can
operate in the two models b and c, for in both cases the
subphotospheric layers will be strongly convective, resulting in the
formation of a corona with variable heating. Using Eq. (1) to estimate
the characteristic period of the fluctuations along with the standard
model for a supermassive star [18] we find that for a mass
$M = 10^8 M_\odot$ and a radius $100 R_{\rm g}$
($R_g = 2GM/c^2$ is the gravitational radius), the period
corresponding to $\omega_c$
would be $\approx 160$ days, a value consistent with observation. On the
accretion disk model, such a period would prevail in the zone of
maximum energy production [9] for a black hole of $ M = 10^8  M_\odot$,
if the luminosity $L = 0.1 L_c$. In galaxy nuclei the fast
component typically exhibits an
approximately constant period in conjunction with sharp changes in
phase [15], a behavior which, it would seem, accords with a convective
wave origin for such fluctuations.

It is worth emphasizing, however, that both in the supermassive
star model and in the model of an accretion disk around a
supermassive black hole, the ratio of
the radiation pressure to the gas pressure is far higher than in the case
of accretion onto a black hole of stellar mass. Acoustic waves will
therefore be damped much more strongly. Our calculations indicate [9]
that in this event the emergent acoustic flux will comprise no more
than 1$\%$ of the flux generated at large optical depth - well below the
variability amplitude observed. The supermassive star model would be
supported if strictly periodic brightness oscillations of constant phase
were detected, associated with the rotation or with pulsations of such a
star as a whole, as might be the case [14] for NGC 4151. But analysis of
observations of several variable nuclei does not reveal any strict
periodicities [19]. Most likely the observed variability should be modeled
by a random process [20] (see, however, Ozernoi et al. [21]).
 The model of a
dense star cluster with frequent supernova outbursts offers the best fit,
in our opinion, to the irregular variability of galaxy nuclei.

\bigskip
\hrule
\bigskip

1. E. Boldt, ``X-ray signatures: new time scales and spectral features," in:
Eighth Texas Sympos. on Relativistic Astrophysics. Ann. New York Acad.
Sci. {\bf 302}, 329-348 (1977).

2. R. A. Syunyaev, ``Variability of X-rays from black holes with accretion disks,"
Astron.Zh. {\bf 49}, 1153-57 (1972) [Sov. Astron. {\bf 16}, 941-944 (1973)].

3. A. P. Lightman and D. M. Eardley, ``Black holes in binary systems:
instability of disk accretion," Astrophys. J. {\bf 187}, L1-L3 (1974).

4. N. Shibazaki and R. Hoshi. ``Structure and stability of an accretion disk around a black
hole," Progr. Theor. Phys. {\bf 54}, 706-718 (1975).

5. N. I. Shakura and R. A. Syunyaev, ``A theory of the instability of disk
accretion onto black holes," Mon. Not. R. Astron. Soc. {\bf 175}, 613-632 (1976).

6. G. S. Bisnovatyi-Kogan and S. I. Blinnikov, ``A hot corona around a black-hole accretion
disk as a model for Cyg X-l," Pis'ma Astron. Zh. {\bf 2}, 489-493 (1976) [Sov. Astron Lett.
{\bf 2}, 191-193 (1977)].

7. G. S. Bisnovatyi-Kogan and S. I. Blinnikov, ``Disk accretion onto a black
hole at subcritical luminosity," Astron. Astrophys. {\bf 59}, 111-125 (1977).

8. M. C. Weisskopf and P. G. Sutherland, ``On the physical reality of the
millisecond bursts in Cyg X-l," Astrophys. J. {\bf 221}, 228-233 (1978).

9. G. S. Bisnovatyi-Kogan and S. I. Blinnikov, ``Wave propagation in a
medium with high radiation pressure" [in Russian]. Preprint Inst. Kosmich.
Issled. Akad. Nauk SSSR No. 421 (1978).

\noindent
[Published in Astrophysics  {\bf 14}, 316-325 (1978); {\bf 15}, 99-107 (1979).]

10. N. I. Shakura and R. A. Syunyaev. ``Black
holes in binary systems: observational appearance," Astron. Astrophys. {\bf 24},
337-355 (1973).

11. N. I. Shakura, R. A. Syunyaev, and S. S. Zilitinkevicn, ``Turbulent
energy transport in accretion disks," Astron. Astrophys. {\bf 62}, 179-187 (1978).

12. J. F. Dolan, C. J. Crannell, B. R. Dennis, K. J. Frost, and L. E. Orwig,
``Intensity transitions in Cyg X-l observed at high energies from OSO-8,"
Nature {\bf 267}, 813-815 (1977).

13. V. M. Lyutyi and A. M. Cherepashchuk, ``H$\alpha$ emission variability in
the Seyfert nuclei NGC 4151, 3516, 1068" [in Russian], Astron. Tsirk.
No. 831. 1-3 (1974).

14. E. T. Belokon', M. K. Babadzhanyants, and V. M. Lyutyi,
``Periodicity of the Seyfert galaxy NGC 4151 in the optical," Astron. Astrophys. Suppl.
{\bf 31}, 383 (1978).

l5. V. M. Lyutyi, ``Optical variability of OJ 287" [in Russian]. Peremennye
Zvezdy {\bf 20}, 243-249 (1976).

16. V. L. Ginzburg and L. M. Ozernoi, ``The nature of quasars
and active galaxy nuclei," Astrophys. Space Sci. {\bf 50}, 23-41 (1977).

17. M. J. Rees, ``Quasar theories,"
in: Eighth Texas Sympos. on Relativistic
Astrophysics. Ann. New York Acad. Sci. {\bf 302}, 613-636 (1977).

18. Ya. B. Zel'dovich and I. D. Novikov, Relativistic Astrophysics, Vol. 1,
Stars and Relativity, Univ. Chicago Press (1971).

19. M. M. Basko and V. M. Lyutyi,
``Search for periodic optical variations of the Seyfert nuclei NGC 1275 and 3516,"
Pis'ma Astron.Zh. {\bf 3}, 104-103 (1977) [Sov. Astron. Lett. {\bf 3}, 54-56 (1977)].

20. W. H. Press, ``Flicker noises in astronomy and elsewhere," Comments
Astrophys. Space Phys. {\bf 7}, 103-119 (1978).

21. L. M. Ozernoi, V. E. Chertoprud, and L. I. Gudzenko, ``Comments on
the light curve of the quasar 3C 273," Astrophys. J. {\bf 216}, 237-243 (1977).

\end{document}
